12.1.1 Observed data

• The average complication rate for liver donor surgeries is about 10%.
• The clients of a particular consultant had only 3 complications in 62 surgeries.
  • \(\hat{p} = 3/62 \approx 0.048\)

12.1.2 Variability of the statistic

• We want to know the actual probability of a complication for the clients of this consultant:
  • parameter: \(p\)
• Our best estimate of \(p\) is the statistic \(\hat{p} \approx 0.048\).
  • But this is just one sample. How would our estimate differ if we took another sample?
• Bootstrapping: We can approximate the variability in different samples by taking samples from a population that looks like an “infinite” version of the original sample data.

Sampling with replacement

• In practice, we sample from the original data with replacement.
• This means that when we select an observation, we add it to the “resample” and then put it back in the original data (we replace it).
• We resample multiple times and obtain many different resampled proportions.
• A dot plot of these bootstrapped proportions approximates a distribution that shows how the statistic varies.

Bootstrap 95% confidence interval

• 95% of the bootstrapped proportions are between 0 and 0.113.
• If you repeated this study many times, 95% of the observed proportions would fall between 0 and 0.113.
  • Interpretation: We are “95% confident” that the actual proportion is between 0 and 0.113.
  • Based on the consultant’s data, it is plausible that the actual proportion of complications with her patients is 0.10. This number is within the range of “typical” values: from 0 to 0.113.

Observed Data

• Consider a bag of 40 Reeses Pieces as a random sample from the population of all Reeses Pieces.
• You obtain a bag of 40 Reeses Pieces and discover that 20 are orange. (\(\hat{p} = 0.5\)).

Variability of the statistic

• Your bag of 40 Reeses Pieces had 20 that were orange. (\(\hat{p} = 0.5\))
• Consider an infinite candy machine containing Reeses Pieces, 50% of which are orange
• We can construct a bootstrap sample by drawing 40 Reeses Pieces out of this machine.
• Each bootstrap sample has a \(\hat{p}_\text{boot}\). Keep track of these in a dotplot to see how the sample proportion varies.

Reeses Pieces Applet

https://www.rossmanchance.com/applets/2021/oneprop/OneProp.htm?candy=1

Group Activity

1. Using the applet, set the Probability of orange to 0.5, the Number of candies to 40, the Number of samples to 1, and choose the Proportion of orange statistic.

2. Click Draw Samples. How many orange did you get? What is \(\hat{p}_\text{boot}\)? Does it equal \(\hat{p}\)? Is it close?

3. Now change Number of samples to 99 and click Draw Samples again. When it finishes, make sure Total Samples is 100.

4. The dotplot that appears should have 100 dots. Use this dotplot to estimate a bootstrap 90% confidence interval for the proportion of orange Reeses Pieces.

12.2.1 Observed data

In a sample of 120 listeners, only 3 were able to correctly guess the tune that was being tapped. (\(\hat{p} = 0.025\))

12.2.2 Variability of the statistic

Simulation:

• Get a bag of 120 marbles, 3 of which are red, and the rest white.
• Draw from the bag 120 times, replacing the marble each time, and keep track of the number of red.
• Compute \(\hat{p}_\text{boot}\)
• Repeat 10000 times

\[ 0.0417 \quad 0.025 \quad 0.025 \quad 0.0083 \quad 0.05 \quad 0.0333 \quad 0.025 \quad 0 \quad 0.0083 \quad 0 \]

Group Activity

1. Reload the applet, set the Probability of orange to 0.5, the Number of candies to 200, and choose the Proportion of orange statistic. You probably also want to uncheck Show animation. Now change Number of samples to 100 and click Draw Samples again. When it finishes, make sure Total Samples is 100.

2. Use this dotplot to estimate a bootstrap 90% confidence interval for the proportion of orange Reeses Pieces.

3. Compare your confidence interval from #2 with your confidence interval from the last activity. How does changing the sample size affect the confidence interval?

Group Activity (continued)

4. Now use the same plot of bootstrapped \(\hat{p}_\text{boot}\)’s to estimate an 80% confidence interval. How does changing the confidence level change the confidence interval?

5. Experiment with some different values for Probability of orange. How does changing the observed \(\hat{p}\) affect the confidence interval?

12.3.1 Plausible range of values for the population parameter

• Given a sample of data, we can construct a confidence interval for \(p\).
  • The center of this interval will be at \(\hat{p}\).
• This confidence interval is a range of plausible values for the population parameter.
  • Based on our data, it is plausible that the unknown parameter lies somewhere in our confidence interval.

12.3.2 Bootstrap confidence interval

• The 95% bootstrap confidence interval for the parameter \(p\) can be obtained directly using the ordered \(\hat{p}_\text{boot}\) values.
• Consider the sorted \(\hat{p}_\text{boot}\) values. Call the 2.5% bootstrapped proportion value “lower,” and call the 97.5% bootstrapped proportion value “upper.”
• The 95% confidence interval is given by: (lower, upper)
• We can use different confidence levels other than 95%. The larger the confidence level, the wider the interval.
• Larger samples give narrower confidence intervals, all else being equal.
• The bootstrap confidence interval will usually be centered on the observed value of \(\hat{p}\).